What are the Eigenvalues of a Sum of (Non-Commuting) Random Symmetric Matrices? : A "Quantum Information" Inspired Answer.

Alan EdelmanRamis Movassagh

July 14, 2011FOCM Random Matrices

Example Resultp=1 classical probabilityp=0 isotropic convolution (finite free probability)

We call this “isotropic entanglement”

Simple Question

The eigenvalues of

where the diagonals are random, and randomly ordered. Too easy?

Another Question

where Q is orthogonal with Haar measure. (Infinite limit = Free probability)

The eigenvalues of

T

Quantum Information Question

where Q is somewhat complicated. (This is the general sum of two symmetric matrices)

The eigenvalues of

T

Preview to the Quantum Information Problem

mxm nxn mxm nxn

Summands commute, eigenvalues addIf A and B are random eigenvalues are classical sum of random variables

Closer to the true problem

d2xd2 dxd dxd d2xd2

Nothing commutes, eigenvalues non-trivial

Actual Problem Hardness =(QMA complete)

di-1xdi-1 d2xd2 dN-i-1xdN-i-1

The Random matrix could be Wishart, Gaussian Ensemble, etc (Ind Haar Eigenvectors)The big matrix is dNxdN

Interesting Quantum Many Body System Phenomena tied to this overlap!

Intuition on the eigenvectors

Classical Quantum Isostropic

Kronecker Product of Haar Measures for A and B

Moments?

Matching Three Moments Theorem

A first try:Ramis “Quantum Agony”

The Departure Theorem

A “Pattern Match”

Hardest to Analyze

The Istropically Entangled Approximation

But this one is hard

The kurtosis

The convolutions

• Assume A,B diagonal. Symmetrized ordering.

• A+B:

• A+Q’BQ:

• A+Qq’BQq

(“hats” indicate joint density is being used)

The Slider Theorem

p only depends on the eigenvectors! Not the eigenvalues

Wishart

Summary